高中生统计推理思维结构的研究
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摘要
统计推理是国际公认的统计学习目标之一,在我国新课程中也受到了一定的重视,但是关于统计推理的研究在我国还非常少。本研究首先翻译梳理了国外对于统计推理的研究,包括对统计推理概念的界定、正确和错误的统计推理类型和评价方式,特别是应用最广泛的团体评价SRA。在梳理文献的过程中,笔者发现目前的团体评价没有揭示学生思维结构的不同层次,于是结合SOLO分类理论和SRA,编制了统计推理调查表。通过对北京地区两个层次五所学校246名高中生的测试,从预设的描述性SOLO分类水平和根据Rasch模型的数量化水平两方面进行数据分析,得到了以下结论:
高中生在统计推理容易出现以下错误观念。在统计量方面,高中生计算均值时容易不考虑极端数据的情况;把平均数与众数、中位数混淆。在统计图方面,他们容易错误选用统计图;从视觉上直观感觉统计图。在抽样方法方面,他们认为好的抽样必须占总体的大比率;只有两小组成员的数目相同时才能比较。在可能性推理方面,学生容易使用概率的直觉模型,这导致他们在做是非判断时只关注单个事件而非事件序列;学生认为小样本与其所在的总体应该相似,所以更喜欢把小样本作为推断和概括的根据;学生估计事件的可能性只取决于它和总体的相似度(例如:认为连续扔硬币,正反交替出现的可能性要比正多反少的可能性大);他们容易出现等可能偏见,把不等机会的事件看成发生机会相同。
本研究还对高中生统计推理思维结构的不同层次作了描述。处于单一结构水平的高中生,不能判断数据类型,认为相关性导致因果;容易受到统计图视觉上的直观误导,可以认识折线图的作用。处于多元结构水平的高中生,会计算平均数,在简单的情境中可以排除极端值的干扰,但在该使用众数、中位数的情况下,容易误用平均数;在描述具体数据时,他们不仅仅从统计图直观上作出推断,而会认真分析统计图的数量关系,得出正确的结论,但在几个统计图中可能会做出不适当的选择;理解概率值为事件发生的可能性大小,但倾向于认为概率大的事件就应该发生,可以在简单的情境中估计事件的可能性大小,但在重复实验或实验序列中,估计事件的可能性只取决于它和总体的相似度;知道使用样本频率推断总体概率,理解大样本的重要性,对于基于小样本的推断采取小心谨慎的态度;理解相关性和因果关系的区别;对随机抽样理解不深刻,担心会偶然的造成两组的不平均。处于关联水平的高中生,会在重复实验序列中判断事件的可能性和不可能性,并初步具备样本空间的意识和概念;理解抽样的可变性,理解抽样过程以及了解大、小样本的效果,能推断出小样本更有可能违反总体的特征。
根据上面的研究结果,本研究讨论了统计教学的建议:教师应在教学中强调、纠正学生的错误观念;教学应着力于提高学生的统计推理思维层次;要尽可能多地给学生体验统计过程的机会;尽可能多地让学生接触实际案例。
Statistical reasoning is one of the statistical learning objectives which received internationally recognition. It has also been attention in the country's new curricula, but the research of statistics reasoning in China is still very small. I translate and card the study of statistical reasoning abroad first, including the definition, the correct and incorrect types and evaluation assessment of statistical reasoning, particularly the most extensive instrument SRA. In the process of carding the literature, I found that the current assessment for large-group evaluation didn’t reveal the structure of different levels, so I worked out the“Questionnaire of statistical reasoning”with SOLO taxonomy and the SRA. Through the testing of 246 students of 5 high school in Beijing, and data analysis from the SOLO descriptive level and the Quantitative level of Rasch model, I draw the following conclusions:
There are some incorrect types statistical reasoning of high school students. In the statistic, most high school students tend to not consider outliers when calculating mean, and mistaken the mean, median and mode. In the charts, they easily choose the wrong figures, and comprehend the charts from the visually intuitive. In the sampling method, they believe that good samples have to represent a high percentage of the population, groups can only be compared if they are the same size. In the reasoning of possibility, an intuitive model of probability that leads students to make yes or no decisions about single events rather than looking at the series of events. They believe that samples should resemble the populations from which they are sampled, which led them to use small samples for making inferences and generalizations about populations. Some students also have the equiprobability bias, and different outcomes of an experiment tend to be viewed as equally likely.
In this study, high school students at different levels of statistical reasoning are described. High school students at the uni-stuctural level can not judge data types, they believe that correlation implies causation, and they vulnerable to the figures intuitive visual misleading. Students at the multi-structural level can calculate mean, consider outliners in easy situations, but in the use of the mode and the median circumstances, easily misuse the mean. In the description of the specific data, they may make inappropriate choice in several statistical figure. They don’t only reason from the intuition, but seriously figures on the relations of number, arrive at a correct conclusion. They understand that probability is the possibility of the incident, but tend to believe that the incident with big probability should occur. They can estimate the possibility in simple cases, but in the experimental sequence, they estimate the possibility only depends on the similarity with the population. They can infer the possibility according to the sample frequency. They understand the importance of big sample, knowing to be skeptical of inferences made using small or biased samples. They understand the difference between correlation and causation. They don’t understand the random sampling clearly, worrying that the result will be uneven because two groups are occasionally divided. Students at the Relational level can judge the possibility and impossibility of events of repeat sequence, and they have the initial awareness and concept of sample space. They understand the variability of sampling, sampling process, as well as the effect of big samples and small samples. They can infer that the small samples are more likely to breach the population characteristics.
According to the findings above, the study discussed the recommendations of the statistics teaching: teachers should rectify the students’misconceptions while teaching; teaching should be concentrated on enhancing statistical reasoning level of thinking; teachers should give students as many opportunities as possible to experience statistical process; students are exposed to actual cases as far as possible.
高中生在统计推理容易出现以下错误观念。在统计量方面,高中生计算均值时容易不考虑极端数据的情况;把平均数与众数、中位数混淆。在统计图方面,他们容易错误选用统计图;从视觉上直观感觉统计图。在抽样方法方面,他们认为好的抽样必须占总体的大比率;只有两小组成员的数目相同时才能比较。在可能性推理方面,学生容易使用概率的直觉模型,这导致他们在做是非判断时只关注单个事件而非事件序列;学生认为小样本与其所在的总体应该相似,所以更喜欢把小样本作为推断和概括的根据;学生估计事件的可能性只取决于它和总体的相似度(例如:认为连续扔硬币,正反交替出现的可能性要比正多反少的可能性大);他们容易出现等可能偏见,把不等机会的事件看成发生机会相同。
本研究还对高中生统计推理思维结构的不同层次作了描述。处于单一结构水平的高中生,不能判断数据类型,认为相关性导致因果;容易受到统计图视觉上的直观误导,可以认识折线图的作用。处于多元结构水平的高中生,会计算平均数,在简单的情境中可以排除极端值的干扰,但在该使用众数、中位数的情况下,容易误用平均数;在描述具体数据时,他们不仅仅从统计图直观上作出推断,而会认真分析统计图的数量关系,得出正确的结论,但在几个统计图中可能会做出不适当的选择;理解概率值为事件发生的可能性大小,但倾向于认为概率大的事件就应该发生,可以在简单的情境中估计事件的可能性大小,但在重复实验或实验序列中,估计事件的可能性只取决于它和总体的相似度;知道使用样本频率推断总体概率,理解大样本的重要性,对于基于小样本的推断采取小心谨慎的态度;理解相关性和因果关系的区别;对随机抽样理解不深刻,担心会偶然的造成两组的不平均。处于关联水平的高中生,会在重复实验序列中判断事件的可能性和不可能性,并初步具备样本空间的意识和概念;理解抽样的可变性,理解抽样过程以及了解大、小样本的效果,能推断出小样本更有可能违反总体的特征。
根据上面的研究结果,本研究讨论了统计教学的建议:教师应在教学中强调、纠正学生的错误观念;教学应着力于提高学生的统计推理思维层次;要尽可能多地给学生体验统计过程的机会;尽可能多地让学生接触实际案例。
Statistical reasoning is one of the statistical learning objectives which received internationally recognition. It has also been attention in the country's new curricula, but the research of statistics reasoning in China is still very small. I translate and card the study of statistical reasoning abroad first, including the definition, the correct and incorrect types and evaluation assessment of statistical reasoning, particularly the most extensive instrument SRA. In the process of carding the literature, I found that the current assessment for large-group evaluation didn’t reveal the structure of different levels, so I worked out the“Questionnaire of statistical reasoning”with SOLO taxonomy and the SRA. Through the testing of 246 students of 5 high school in Beijing, and data analysis from the SOLO descriptive level and the Quantitative level of Rasch model, I draw the following conclusions:
There are some incorrect types statistical reasoning of high school students. In the statistic, most high school students tend to not consider outliers when calculating mean, and mistaken the mean, median and mode. In the charts, they easily choose the wrong figures, and comprehend the charts from the visually intuitive. In the sampling method, they believe that good samples have to represent a high percentage of the population, groups can only be compared if they are the same size. In the reasoning of possibility, an intuitive model of probability that leads students to make yes or no decisions about single events rather than looking at the series of events. They believe that samples should resemble the populations from which they are sampled, which led them to use small samples for making inferences and generalizations about populations. Some students also have the equiprobability bias, and different outcomes of an experiment tend to be viewed as equally likely.
In this study, high school students at different levels of statistical reasoning are described. High school students at the uni-stuctural level can not judge data types, they believe that correlation implies causation, and they vulnerable to the figures intuitive visual misleading. Students at the multi-structural level can calculate mean, consider outliners in easy situations, but in the use of the mode and the median circumstances, easily misuse the mean. In the description of the specific data, they may make inappropriate choice in several statistical figure. They don’t only reason from the intuition, but seriously figures on the relations of number, arrive at a correct conclusion. They understand that probability is the possibility of the incident, but tend to believe that the incident with big probability should occur. They can estimate the possibility in simple cases, but in the experimental sequence, they estimate the possibility only depends on the similarity with the population. They can infer the possibility according to the sample frequency. They understand the importance of big sample, knowing to be skeptical of inferences made using small or biased samples. They understand the difference between correlation and causation. They don’t understand the random sampling clearly, worrying that the result will be uneven because two groups are occasionally divided. Students at the Relational level can judge the possibility and impossibility of events of repeat sequence, and they have the initial awareness and concept of sample space. They understand the variability of sampling, sampling process, as well as the effect of big samples and small samples. They can infer that the small samples are more likely to breach the population characteristics.
According to the findings above, the study discussed the recommendations of the statistics teaching: teachers should rectify the students’misconceptions while teaching; teaching should be concentrated on enhancing statistical reasoning level of thinking; teachers should give students as many opportunities as possible to experience statistical process; students are exposed to actual cases as far as possible.
引文
[1] Gal. I., & Garfield, J. (Eds.). The assessment challenge in statistics education [M/OL]. Amsterdam: IOS Press, 1997. http://www.stat.auckland.ac.nz/~iase/publications/assessbk/chapter01.pdf
[2] Dirk Tempelaar. Statistical Reasoning Assessment: an Analysis of the SRA Instrument [EB/OL].Paper presented at the ARTIST Roundtable Conference on Assessment in Statistics held at Lawrence University, August 1-4, 2004. http://www.rossmanchance.com/artist/proceedings/tempelaar.pdf
[3] 数学课程标准研制组. 全日制义务教育数学课程标准(实验)解读[M]. 北京: 北京师范大学出版社, 2002 年 5 月第 1 版.
[4] 高中数学课程标准研制组. 普通高中数学课程标准(实验)解读[M]. 南京: 江苏教育出版社, 2004 年 4月第 1 版.
[5] Chervaney, N. Collier, R. Fienberg, S., Johnson, P, and Neter, J. A framework for the development of measurement instruments for evaluating the introductory statistics course [J]. The American Statistician, 1977, 31, 17-23.
[6] Chervaney, N. Benson, P.G., and Iyer, R. The planning stage in statistical reasoning [J]. The American Statistician, 1980, 34, 222-226.
[7] Marsha Lovett. A Collaborative Convergence on Studying Reasoning Processes: A Case Study in Statistics[C/OL]. To appear in D. Klahr & S. Carver (Eds.) Cognition and Instruction: 25 Years of Progress. Mahwah, NJ: Erlbaum,2001. http://www.psy.cmu.edu/LAPS/pubs/Lovett01CandI.pdf
[8] Garfield, J., and Gal. I. Teaching and Assessing Statistical Reasoning [C]. In Developing Mathematical Reasoning in Grades K-12, ed. L. Stiff, Reston, VA: National Council Teachers of Mathematics, 1999, 207-219.
[9] Joan Garfield, Robert delMas. The Web-based ARTIST: Assessment Resource Tools for Improving Statistical Thinking [EB/OL]. Paper presented in the Symposium: Assessment of Statistical Reasoning to Enhance Educational Quality AERA Annual Meeting, Chicago, April 2003. https://app.gen.umn.edu/artist/articles/AERA_2003.pdf
[10] Reija Helenius. Co-operation with Educational Institutions: a Strategic Challenge for Statistical Offices [J/OL]. International Statistical Institute, 55th Session 2005. http://www.stat.auckland.ac.nz/~iase/publications/13/Helenius.pdf
[11] Rumsey, D. J. Statistical Literacy as a Goal for Introductory Statistics Courses [J/OL]. Journal of Statistics Education, 2002, 10(3). http://www.amstat.org/publications/jse/v10n3/rumsey2.html
[12] Chance, B. L. Components of Statistical Thinking and Implications for Instruction and Assessment [J/OL]. Journal of Statistics Education, 2002,10(3). http://www.amstat.org/publications/jse/v10n3/rumsey2.html
[13] Wild, C. J., Pfannkuch, M. Statistical Thinking in Empirical Enquiry [J/OL]. International Statistical Review, 1999, 67, 223-265. http://www.stat.auckland.ac.nz/~iase/publications/isr/99.WP.pdf
[14] [26] Robert C. delMas. Statistical Literacy, Reasoning, and Learning: A Commentary [J/OL]. Journal of Statistics Education 2002, 10(3). http://www.amstat.org/publications/jse/v10n3/delmas_discussion.html
[15] American Association for the Advancement of Science. Benchmarks for Science Literacy[M]. New York: Oxford University Press,1993.
[16] [21] [22] [23] Kahneman, D., Slovic, P. and Tversky, A. Judgment Under Uncertainty: Heuristics and Biases [M/OL]. Cambridge, UK: Cambridge University Press, 1982. http://www.garfield.library.upenn.edu/classics1983/A1983QG78800001.pdf
[17] Garfield, J., and Ahlgren, A. Difficulties in learning basic concepts in probability and statistics: implications for research[J/OL]. Journal for Research in Mathematics Education, 1988, 19, 44-63. http://www.jstor.org/view/00218251/ap020087/02a00040/0
[18] [20] Konold, C. Informal conceptions of probability [J/OL]. Cognition and Instruction, 1989, 6, 59-98. http://www.jstor.org/view/07370008/sp050020/05x0139i/0
[19] [24] Lecoutre, M. P. Cognitive models and problem spaces in “purely random” situations [J/OL]. Educational Studies in Mathematics, 1992, 23, 557-568. http://www.springerlink.com/index/J607P445VL683633.pdf
[25] Garfield, J., and Chance, B. Assessment in Statistics Education: Issues and Challenges [J/OL]. Mathematics Thinking and Learning, 2000, 2, 99-125. https://app.gen.umn.edu/artist/articles/Garfield02.pdf
[27] Hawkins, A. , Jolliffe, F. , Glickman, L. Teaching statistical concepts [M]. London:Longman, 1992.
[28] Colvin, S. , Vos, K. Authentic assessment models for statistics education[C]. In I. Gal & J. B. Garfield (Eds.), The assessment challenge in statistics education. Amsterdam: IOS Press, 1997, 27–36.
[29] Lesh,R. , Amit,M., Schorr,R. Using “real-life” problems to prompt students to construct conceptual models for statistical reasoning[C/OL]. In I. Gal & J.B.Garfield (Eds.), The assessment challenge in statistics education. Amsterdam: IOS Press, 1997, 65–83. http://www.stat.auckland.ac.nz/~iase/publications/assessbk/chapter06.pdf
[30] Keeler, C. Portfolio assessment in graduate level statistics courses [M/OL]. In I.Gal&J.B.Garfield (Eds.), The assessment challenge in statistics education. Amsterdam: IOS Press, 1997, 165–178. http://www.stat.auckland.ac.nz/~iase/publications/assessbk/chapter13.pdf
[31] Schau, C., Mattern, N. Assessing students’connected understanding of statistical relationships[M/OL]. In I.Gal, J.B.Garfield (Eds.), The assessment challenge in statistics education. Amsterdam: IOS Press, 1997, 91–104. https://app.gen.umn.edu/artist/articles/Schau.pdf
[32] Watson, J. Assessing statistical thinking using the media [M/OL]. In I.Gal & J.B.Garfield (Eds.), The assessment challenge in statistics education. Amsterdam: IOS Press, 1997, 107–121. http://www.stat.auckland.ac.nz/~iase/publications/assessbk/chapter09.pdf
[33] Angelo, T., Cross, K.P. Classroom assessment techniques [M/OL]. San Francisco: Jossey-Bass, 1993. http://honolulu.hawaii.edu/intranet/committees/FacDevCom/guidebk/teachtip/assess-1.htm
[34] Cicchitelli, G., Bartolucci, F., Forcina, A.. Assessment in statistics using the personal computer [EB/OL]. Paper presented at the 52nd International Statistical Institute Session, Helsinki, Finland, 1999, August. http://www.stat.auckland.ac.nz/~iase/publications/5/cicc0349.pdf
[35] Garfield, J. . Challenges in assessing statistical reasoning [EB/OL]. Paper presented at the annual meeting of the American Educational Research Association, San Diego, CA, 1998,April.
[36] HUI-JU C.LIU, JOAN B.GARFIELD. Sex Differences in Statistical Reasoning [J]. 台湾师范大学教育心理与辅导学系教育心理学报, 2002, 34 卷, 1 期, 123-138.
[37] 王文芳. 高中生统计推理的现状调查与比较研究[D]. 东北师范大学, 2006, 硕士论文.
[38] Biggs, J . B. , & Collis , K. F. . Evaluating the Quality of Learning: the SOLO Taxonomy[M]. New York: Academic Press. 1982.
[39] [42]张春莉, 石文静. SOLO 分类理论在数学评价与教学中的应用[J]. 小学数学教育, 2007, 第 3 期.
[40] 李祥兆. 学生思维评价的新视角——SOLO 分类评价理论评述[J]. 教育科学研究, 2005, 11.
[41] 蔡永红. SOLO 分类理论及其在教学中的应用[J]. 教师教育研究, 2006, 18 (1).
[43] 李俊. 中小学概率的教与学[M]. 上海: 华东师范大学出版社, 2003 年 5 月第一版, 100.
[2] Dirk Tempelaar. Statistical Reasoning Assessment: an Analysis of the SRA Instrument [EB/OL].Paper presented at the ARTIST Roundtable Conference on Assessment in Statistics held at Lawrence University, August 1-4, 2004. http://www.rossmanchance.com/artist/proceedings/tempelaar.pdf
[3] 数学课程标准研制组. 全日制义务教育数学课程标准(实验)解读[M]. 北京: 北京师范大学出版社, 2002 年 5 月第 1 版.
[4] 高中数学课程标准研制组. 普通高中数学课程标准(实验)解读[M]. 南京: 江苏教育出版社, 2004 年 4月第 1 版.
[5] Chervaney, N. Collier, R. Fienberg, S., Johnson, P, and Neter, J. A framework for the development of measurement instruments for evaluating the introductory statistics course [J]. The American Statistician, 1977, 31, 17-23.
[6] Chervaney, N. Benson, P.G., and Iyer, R. The planning stage in statistical reasoning [J]. The American Statistician, 1980, 34, 222-226.
[7] Marsha Lovett. A Collaborative Convergence on Studying Reasoning Processes: A Case Study in Statistics[C/OL]. To appear in D. Klahr & S. Carver (Eds.) Cognition and Instruction: 25 Years of Progress. Mahwah, NJ: Erlbaum,2001. http://www.psy.cmu.edu/LAPS/pubs/Lovett01CandI.pdf
[8] Garfield, J., and Gal. I. Teaching and Assessing Statistical Reasoning [C]. In Developing Mathematical Reasoning in Grades K-12, ed. L. Stiff, Reston, VA: National Council Teachers of Mathematics, 1999, 207-219.
[9] Joan Garfield, Robert delMas. The Web-based ARTIST: Assessment Resource Tools for Improving Statistical Thinking [EB/OL]. Paper presented in the Symposium: Assessment of Statistical Reasoning to Enhance Educational Quality AERA Annual Meeting, Chicago, April 2003. https://app.gen.umn.edu/artist/articles/AERA_2003.pdf
[10] Reija Helenius. Co-operation with Educational Institutions: a Strategic Challenge for Statistical Offices [J/OL]. International Statistical Institute, 55th Session 2005. http://www.stat.auckland.ac.nz/~iase/publications/13/Helenius.pdf
[11] Rumsey, D. J. Statistical Literacy as a Goal for Introductory Statistics Courses [J/OL]. Journal of Statistics Education, 2002, 10(3). http://www.amstat.org/publications/jse/v10n3/rumsey2.html
[12] Chance, B. L. Components of Statistical Thinking and Implications for Instruction and Assessment [J/OL]. Journal of Statistics Education, 2002,10(3). http://www.amstat.org/publications/jse/v10n3/rumsey2.html
[13] Wild, C. J., Pfannkuch, M. Statistical Thinking in Empirical Enquiry [J/OL]. International Statistical Review, 1999, 67, 223-265. http://www.stat.auckland.ac.nz/~iase/publications/isr/99.WP.pdf
[14] [26] Robert C. delMas. Statistical Literacy, Reasoning, and Learning: A Commentary [J/OL]. Journal of Statistics Education 2002, 10(3). http://www.amstat.org/publications/jse/v10n3/delmas_discussion.html
[15] American Association for the Advancement of Science. Benchmarks for Science Literacy[M]. New York: Oxford University Press,1993.
[16] [21] [22] [23] Kahneman, D., Slovic, P. and Tversky, A. Judgment Under Uncertainty: Heuristics and Biases [M/OL]. Cambridge, UK: Cambridge University Press, 1982. http://www.garfield.library.upenn.edu/classics1983/A1983QG78800001.pdf
[17] Garfield, J., and Ahlgren, A. Difficulties in learning basic concepts in probability and statistics: implications for research[J/OL]. Journal for Research in Mathematics Education, 1988, 19, 44-63. http://www.jstor.org/view/00218251/ap020087/02a00040/0
[18] [20] Konold, C. Informal conceptions of probability [J/OL]. Cognition and Instruction, 1989, 6, 59-98. http://www.jstor.org/view/07370008/sp050020/05x0139i/0
[19] [24] Lecoutre, M. P. Cognitive models and problem spaces in “purely random” situations [J/OL]. Educational Studies in Mathematics, 1992, 23, 557-568. http://www.springerlink.com/index/J607P445VL683633.pdf
[25] Garfield, J., and Chance, B. Assessment in Statistics Education: Issues and Challenges [J/OL]. Mathematics Thinking and Learning, 2000, 2, 99-125. https://app.gen.umn.edu/artist/articles/Garfield02.pdf
[27] Hawkins, A. , Jolliffe, F. , Glickman, L. Teaching statistical concepts [M]. London:Longman, 1992.
[28] Colvin, S. , Vos, K. Authentic assessment models for statistics education[C]. In I. Gal & J. B. Garfield (Eds.), The assessment challenge in statistics education. Amsterdam: IOS Press, 1997, 27–36.
[29] Lesh,R. , Amit,M., Schorr,R. Using “real-life” problems to prompt students to construct conceptual models for statistical reasoning[C/OL]. In I. Gal & J.B.Garfield (Eds.), The assessment challenge in statistics education. Amsterdam: IOS Press, 1997, 65–83. http://www.stat.auckland.ac.nz/~iase/publications/assessbk/chapter06.pdf
[30] Keeler, C. Portfolio assessment in graduate level statistics courses [M/OL]. In I.Gal&J.B.Garfield (Eds.), The assessment challenge in statistics education. Amsterdam: IOS Press, 1997, 165–178. http://www.stat.auckland.ac.nz/~iase/publications/assessbk/chapter13.pdf
[31] Schau, C., Mattern, N. Assessing students’connected understanding of statistical relationships[M/OL]. In I.Gal, J.B.Garfield (Eds.), The assessment challenge in statistics education. Amsterdam: IOS Press, 1997, 91–104. https://app.gen.umn.edu/artist/articles/Schau.pdf
[32] Watson, J. Assessing statistical thinking using the media [M/OL]. In I.Gal & J.B.Garfield (Eds.), The assessment challenge in statistics education. Amsterdam: IOS Press, 1997, 107–121. http://www.stat.auckland.ac.nz/~iase/publications/assessbk/chapter09.pdf
[33] Angelo, T., Cross, K.P. Classroom assessment techniques [M/OL]. San Francisco: Jossey-Bass, 1993. http://honolulu.hawaii.edu/intranet/committees/FacDevCom/guidebk/teachtip/assess-1.htm
[34] Cicchitelli, G., Bartolucci, F., Forcina, A.. Assessment in statistics using the personal computer [EB/OL]. Paper presented at the 52nd International Statistical Institute Session, Helsinki, Finland, 1999, August. http://www.stat.auckland.ac.nz/~iase/publications/5/cicc0349.pdf
[35] Garfield, J. . Challenges in assessing statistical reasoning [EB/OL]. Paper presented at the annual meeting of the American Educational Research Association, San Diego, CA, 1998,April.
[36] HUI-JU C.LIU, JOAN B.GARFIELD. Sex Differences in Statistical Reasoning [J]. 台湾师范大学教育心理与辅导学系教育心理学报, 2002, 34 卷, 1 期, 123-138.
[37] 王文芳. 高中生统计推理的现状调查与比较研究[D]. 东北师范大学, 2006, 硕士论文.
[38] Biggs, J . B. , & Collis , K. F. . Evaluating the Quality of Learning: the SOLO Taxonomy[M]. New York: Academic Press. 1982.
[39] [42]张春莉, 石文静. SOLO 分类理论在数学评价与教学中的应用[J]. 小学数学教育, 2007, 第 3 期.
[40] 李祥兆. 学生思维评价的新视角——SOLO 分类评价理论评述[J]. 教育科学研究, 2005, 11.
[41] 蔡永红. SOLO 分类理论及其在教学中的应用[J]. 教师教育研究, 2006, 18 (1).
[43] 李俊. 中小学概率的教与学[M]. 上海: 华东师范大学出版社, 2003 年 5 月第一版, 100.